# Law of Exponents Cheatsheet

math exponents law

I was studying rationals this week so I thought that creating an exponents cheat sheet for future reference is a good idea.

## Powers of 0

$$a^0 = 1$$

Example:

$$3^0 = 1$$

## Powers of 1

$$a^1 = a$$

Example:

$$7^1 = 7$$

## Power of 1 with base 0

$$0^0 = Indeterminate$$

## Multiplying powers with the same base

$$a^{x}a^{y} = a^{x+y}$$
Example:
$$$$\begin{split} a^{2}a^{3} &= (aa)(aaa) \\ &= (aaaaa) \\ &= a^5 \end{split}$$$$
$$$$\begin{split} a^{2}a^{3} &= a^{2+3} \\ &= a^5 \end{split}$$$$

## Dividing powers with the same base

$$\frac{a^{x}}{a^{y}} = a^{x-y}$$

Example:

$$$$\begin{split} a^{5}a^{3} &= (aaaaa)/(aaa) \\ &= aa \\ &= a^2 \end{split}$$$$
$$$$\begin{split} a^{5}a^{3} &= a^{5-3} \\ &= a^2 \end{split}$$$$

## Exponentiation of a power

$$(a^{x})^y = a^{xy}$$

Example:

$$$$\begin{split} (a^2)^3 &= (aa)^3 \\ &= (aa)(aa)(aa) \\ &= (aaaaaa) \\ &= a^6 \end{split}$$$$
$$$$\begin{split} (a^2)^3 &= a^{2 \cdot 3} \\ &= a^6 \end{split}$$$$

## Distributive Property of Exponents

$$(ab)^x = a^{x}b^{x}$$

Example:

$$(2 \cdot 3)^4 = 2^{4} \cdot 3^{4} = 16 \cdot 81 = 1296$$

## Distributive Property of Exponents on Fractions

$$\Bigg(\frac{a}{b}\Bigg)^x = \frac{a^x}{b^x}$$

Example:

$$\Bigg(\frac{7}{9}\Bigg)^4 = \frac{7^4}{9^4} = \frac{2401}{6561}$$

## Negative One Property of Exponents

$$a^{-n} = \frac{1}{a^n}$$

Example:

$$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$

## Nth Root

$$a^{\frac{x}{y}} = \sqrt[y]{a^x}$$

Example:

$$7^{\frac{2}{3}} = \sqrt[3]{7^2} = \sqrt[3]{49}$$